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Recently someone mentioned to me that there is a diophantine equation that looks very simple and innocent, but the smallest solution involves numbers of the order $10^{50}$ or something like this. The equation is probably in either 1,2, or 3 varaibles. It has low coefficients, probably all 1 or 2. And the degree is low also, probably 4 or less.

Is there such an equation?

Edit: I think the equation might have been studied by Fermat, but I'm not sure.

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    $\begingroup$ Contact your friend. I do not see how it is our responsibility to guess what your friend meant, especially if he was not entirely clear on the details. $\endgroup$ – Will Jagy May 30 '15 at 17:25
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    $\begingroup$ The smallest solution of the Pell equation $\ p^2-d q^2 \!= 1,\ \ d = 4\cdot 609\cdot 7766\cdot 4657^2 $ arising from the ancient Archimedes Cattle Problem has over a couple hundred thousand decimal digits. $\endgroup$ – Bill Dubuque May 30 '15 at 17:31
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    $\begingroup$ @BillDubuque: I've seen the Archimedes cattle problem but it is not at all suprising that the solution will be very large, because it is a system with 10 or so equations. $\endgroup$ – Joshua Benabou May 30 '15 at 17:51
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    $\begingroup$ @Joshua The Archimedes cattle problem starts with seven tiny linear Diophantine equations and reduces to solving $\ x^2 - d y^2 = 1\ $ for $\ d = 410286423278424.\ $ You don't find it surprising that the smallest solution has $\,206544\,$ decimal digits? $\endgroup$ – Bill Dubuque May 30 '15 at 17:58
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    $\begingroup$ The most famous versions of Pell´s equation is the above mentioned Archimedes cattle problem and the case d=61 with x=1 766 319 049, y= 226 153 980. $\endgroup$ – Mikael Jensen May 30 '15 at 18:21
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The smallest (in terms of naive height) solution of $y^2=x^3+877x$ is

$$\left(\frac{375494528127162193105504069942092792346201}{6215987776871505425463220780697238044100},\frac{256256267988926809388776834045513089648669153204356603464786949}{490078023219787588959802933995928925096061616470779979261000}\right)$$

This is an example of Bremner and Cassels. Thus, the smallest solution of $ZY^2=X^3+877XZ^2$ is $$(29604565304828237474403861024284371796799791624792913256602210,256256267988926809388776834045513089648669153204356603464786949,490078023219787588959802933995928925096061616470779979261000).$$ The $X$ coordinate is $>2\cdot 10^{61}$.

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Here is a general comment about the mechanism behind this phenomenon. By Matiyasevich's theorem, the problem of determining whether a Diophantine equation has a solution is undecidable. This implies that it is not possible to give a computable a priori bound on the size of the solutions to a Diophantine equation (since, given such a bound, we could solve Diophantine equations by checking all solutions up to the bound), so it follows that the size of the smallest solution to a Diophantine equation eventually exceeds any computable function of the Diophantine equation.

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  • $\begingroup$ For small degrees and symmetric equations is always possible to write a formula for solving equations. Everything solved quadratic forms. $\endgroup$ – individ May 30 '15 at 19:15
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Here's a monster. The smallest integer solution to,

$$(x + a)^7 + (x - a)^7 + (2x + b)^7 + (2x - b)^7 + \\(-x - c)^7 + (-x + c)^7 + (-2x - d)^7 + (-2x + d)^7 \\= 14^7(a^6 + 2b^6 - c^6 - 2d^6)^7$$

has $x$ with $\color{red}{1179\; \text{digits}}$. The variables $a,b,c,d$ are,

$$292565171139318137956759657471297,\\ 863420822620431936290192229011966,\\ 534407060429869176086407612538177,\\ 859793943610761912321826231621886$$

and

$$\small{x =481563304865430516682423843723465575123177045754683810551700\dots \approx \color{red}{4.81 \times 10^{1178}}}$$

Ajai Choudhry found this using an elliptic curve, which may explain the large values.

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